It is easy to see that Julia's age must be a common multiple of $2$, $3$, $4$, $6$ plus one, i.e. $24k + 1$ where $k$ is a positive integer. Meanwhile it must be a multiple of $5$. Because $24$ is an even number, in order for $24k+1$ is a multiple of $5$, $24k$ must end with $4$. Clearly $k=1$ is a solution. $k=6$ is the next solution, but will result in a three-digit number. Hence $k=1$ is the only solution. Julia's age is $25$, Hence, Bart is $\boxed{5}$ year old.